Influence of noise on power-law scaling functions and an algorithm for dimension estimations

The influence of Gaussian noise on power-law scaling functions of interpoint distances has been investigated. These functions appear in the estimation of the correlation dimension $\alpha$ of the attractor of a chaotic dynamical system, where the relative number of pairwise distances smaller than $r$ (correlation integral) theoretically scales as $r^{\alpha }$. Assuming the noise added to each measurement is independent and the distribution of the distances is governed completely by the power-law scaling rule in the noise-free case, the scaling functions of the perturbed distances have been calculated exactly. By considering the limiting cases for small and large distances, a method is presented to estimate the variance of the added noise and approximations of the scaling functions, which are suitable for data analysis, are derived. Dimension estimation can be improved by applying a nonlinear fit procedure to histograms of interpoint distances instead of the usual linear regression on log-log plots.